Hybrid inverse problems

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Abstract

Inverse problems arise in different disciplines including exploration geophysics, medical imaging and nondestructive evaluation. In some settings, a single modality displays either high contrast or high resolution but not both. In favorable situations, physical effects couple one high-contrast modality with another high-resolution modality. Hybrid inverse problems, also called coupled-physics inverse problems or multi-wave inverse problems, are motivated to study these coupling mechanisms to display both high contrast and high resolution. In photo-acoustic tomography(PAT) and thermo-acoustic tomography(TAT), acoustic waves couple with optical radiations, while in electro-seismic(ES) effect, seismic waves couple with electrical fields. The solution strategies of hybrid methods typically consist of two steps. Normally in the first step, a high-resolution-low-contrast modality is considered to reconstruct some internal data. In PAT and TAT, we invert a wave equation and reconstruct the initial wave pressure from available boundary measurements. In ES conversion, we invert Biot's system to recover the internal source. In our work, we assume that this step has been performed. The second step consists of the quantitative reconstruction of coefficients of interest by applying a high-contrast modality on the high-resolution internal data obtained during the first step. In the second step of PAT, our main objective is to recover diffusion and absorption coefficients from the internal measurements. The second step of ES conversion works with Maxwell's equation to reconstruct the conductivity and the coupling coefficient from internal measurements. This thesis mainly focuses on the second steps of PAT and ES conversion. Indeed, assuming the internal measurements are obtained already, we mainly prove the uniqueness and the stability of the constructions of the coefficients of interest in PAT and ES conversion. Precisely, We show that knowledge of two internal data based on well-chosen boundary conditions uniquely determine the coefficients of interest. Moreover, Lipschitz type stability results are proved based on the same sets of well-chosen boundary conditions.